= Solution
Let $a$ be a characteristic crack depth and $Y$ its geometry factor. A simple fast-fracture condition is
$$
K_I=Y\sigma\sqrt{\pi a}=K_{IC},\qquad
a_c=\frac1\pi\left(\frac{K_{IC}}{Y\sigma}\right)^2.
$$
The crack should penetrate a wall of thickness $t$ before reaching the critical size, so the simplified <leak-before-break criterion> is $t<a_c$, with appropriate geometry and stable-through-wall-crack assumptions.
\b[The conventional material-based maximum thickness uses the yield <stress> as a conservative limiting operating <stress>]:
$$
\boxed{t<t_{\max}^{(\sigma_y)}=
\frac1\pi\left(\frac{K_{IC}}{Y\sigma_y}\right)^2.}
$$
It ensures that the depth required for penetration is subcritical even at the stated limiting <stress>. Geometrical factors depend on crack shape and orientation; this is an educational screening estimate rather than a full crack-growth or leak-detection analysis. The handbook's fracture formula uses the same stress-intensity and geometry-factor convention.<Materials Data Book by Cambridge University Engineering Department>[2011 edition revised2019, printed p.6, Fast Fracture; this confirms the formula convention, not the exact2018NST supplied values]{ref}
There is an important distinction if the actual pressure and radius are fixed while thickness varies. Substituting $\sigma=Pr/t$ at penetration $a=t$ gives
$$
K_I(t)=YPr\sqrt{\frac\pi t},
\qquad
\boxed{t>\pi\left(\frac{YPr}{K_{IC}}\right)^2.}
$$
That is a \b[lower], not upper, bound on thickness under the fixed-pressure model. Yield avoidance additionally requires $t\gtrsim Pr/\sigma_y$. It would be inconsistent to derive the fixed-pressure expression and call it a maximum. The standard maximum quoted above answers a different, yield-limited material-selection comparison.
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